From morby@obs-nice.fr Thu Mar 15 16:18:33 2001 Date: Mon, 28 Aug 2000 17:23:06 +0200 (MET DST) From: Alessendro MORBIDELLI To: vokrouhl@cesnet.cz, froig@iagusp.usp.br, bottke@ceres.tn.cornell.edu, david@obs-nice.fr, miroslav.broz@email.cz Subject: a first analysis of the eunomia family Hello these are the preliminary steps that we did on the eunomia family. 1) We run the proper elements code on the integration (136My finished). 2) we plot the evolution in the proper-(a,e) plane all the particles. the plot is attached below. You see several vertical funtains, evident trace of the action of mini resonances. The major funtain at the center of the plot is the 11/4 resonance with Jupiter. 3) because in the integratio we set no--reorientation, if there weren't resonances, all particles would drift in semimajor axis linearly in time. If the semimajor axis stops, inverts its derivative, or changes of drift rate, it is because a resonance is hit. Thus for each particle we can fit the evolution of a(t) with a function A*t+B. A & B are determined by best fit. Then, for each particle, the quantity sigma_fit=\int [a(t)-A*t-B]**2 dt tells us the quality of the linear fit. Particles that drifted linearly with time have a small simga_fit, while those that hit resonances, and thus have a non linear evolution have a large simga_fit. 4) for each particle we also compute Delta e, that is the difference between the maximum and the minimum eccentricity evere reached. 5) finally we plot Delta e as a function of sigma_fit. The plot is attached below. as you can see, there is an evident correlation between Delta e and sigma_fit. that is because the only bodies that drift in e are those captured in the miniresonances and thus have a bad linar fit. The few exceptions (i.e. particles with large Delta e and small sigma_fit) are particles that stay in the resonance all time, so that the linear fit a(t)=B (A=0) is a good fit! In conclusion, the plot shows that in 136 My, roughly 50% of the particles did interact with mini--resonances, leaving a trace in the behavior of a(t) (strongly non--linear) and in the diffusion of e. This is neat. 7) finally, we have focussed on the 11/4 resonance. Several particles interact with this resonance. 4 stay in the resonance all time; 4 are captured in the resonance from above; 6 are captured in the resonance from below; 9 are ejected from the resonance (they were either captured before or were already in at the beginning of the integration); 11 cross the resonance from either direction and do not show any anomaly during the crossing 4 exhibit short captures while crossing the resonance. In conclusion, capture is possible in either direction (violating adiabatic theory, which evidently does not apply in these cases). The capture probability is roughly 50% This goes in the direction of the conjecture of David and Paolo's Science paper. Miniresonances do interact with Yarko--drifting bodies.!!!!!!!!!! This is all for today. More tomorrow on the dispersion of the family as a function of time. Cheers Morby [ Part 2, "eunomia.ps" Application/POSTSCRIPT 1.1MB. ] [ Unable to print this part. ] [ Part 3, "grafic.ps" Application/POSTSCRIPT 25KB. ] [ Unable to print this part. ]